Abstract
In this paper, we study the quaternion matrix equation (1.1) by using the semi-tensor product of matrices. First, we propose a real vector representation of a quaternion matrix and study its properties. Then combining this real vector representation with semi-tensor product of matrices, the explicit expressions of the least squares solution, the least squares J-self-conjugate solution and the least squares anti-J-self-conjugate solution of (1.1) are proposed. We give the equivalence conditions for the compatibility of (1.1) and the expressions of the minimal norm solution with above-mentioned properties. We also propose the corresponding algorithms and give numerical experiments to verify the effectiveness of our methods.
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Communicated by Dietmar Hildenbrand.
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W. Ding: This work was completed with the support of the Natural Science Foundation of Shandong under grant ZR2020MA053.
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Ding, W., Li, Y. & Wang, D. A Real Method for Solving Quaternion Matrix Equation \(\mathbf{X}-\mathbf{A}{\hat{\mathbf{X}}}{} \mathbf{B} = \mathbf{C}\) Based on Semi-Tensor Product of Matrices. Adv. Appl. Clifford Algebras 31, 78 (2021). https://doi.org/10.1007/s00006-021-01180-1
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DOI: https://doi.org/10.1007/s00006-021-01180-1
Keywords
- Quaternion matrix equation
- Least squares solution
- J-self-conjugate matrix
- Anti-J-self-conjugate matrix
- Semi-tensor product of matrices
- Real vector representation